After defining an original zero-point, we set the particle moving, hoping to plot a nice curve detailing the likelyhood of a particle being at any given point on our axis.
Now at any given time, the most likely position after a further time interval is the current position, wherever that may be, not the original zero-point (Agreed?).
We would therefore <expect> the particle do drift off and end up nowhere near the original 0, as in CycloWizard's Matlab program.
Would the overall frequency distribution still be gaussian? I'm fairly sure that the discreteness of this example shouldn't matter...
Good answers though guys, will read your various sources and get back to you all!
Well that's an entirely different situation...look up "conditional probability." What you're doing now is conditioning on some event, which changes the probabilistic universe.
I doubt that makes much sense, so here's an example:
I throw 2 fair coins independently. If they both show heads, I win; else I lose.
Now straight up, the probability of me winning is 1/4 (1/2 * 1/2).
But now I ask...what is the probability of winning *given* that the first coin landed heads up? It's 1/2.
By knowing something about the first coin, it completely changes the probability of me winning.
In the same way, by knowing something about how the first 500 moves turned out, that tells me about the where I can end up in the next 500 moves. For example, if I start at 0 and end up at +100 after 500 moves, then I know it's impossible to reach -500 after the next 500 moves.
So when you try to "reset" the situation in the scenario you describe, you are trying to compare apples & oranges.
Edit: TuxDave, isn't it funny how a lot of mathematical concepts/theorems/etc make perfect sense in 1 & 2D, but as soon as you get to 3D and higher, everything gets crazy? heh it seems like for many things, the jump from 1D to 2D isn't a big deal...but from 2D to 3D can be weird. But from 3D and onward, it's all the same.